论文标题
无需选择的最大设置
Maximal sets without Choice
论文作者
论文摘要
我们表明，相对于ZF是一致的，没有$ \ mathbb {r} $的井井有条，而存在一系列特殊的真实集，例如Hamel Bases，超越基础，Vitali Sets或Bernstein Sets。更确切地说，我们可以假设$ \ mathbb {r} $上的每个投射超图具有最大独立集，其中包括其他一些事情。例如，我们获得了所有投影对等关系的横向。此外，这是可能的，而$ \ mathsf {dc} _ {ω_1} $保留，或者对reals的可数选择失败。假设不可访问的红衣主教的一致性，“投影”甚至可以用“ $ l（\ mathbb {r}）$替换。这极大地增强了早期的一致性导致文献。
We show that it is consistent relative to ZF, that there is no well-ordering of $\mathbb{R}$ while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we can assume that every projective hypergraph on $\mathbb{R}$ has a maximal independent set, among a few other things. For example, we get transversals for all projective equivalence relations. Moreover, this is possible while either $\mathsf{DC}_{ω_1}$ holds, or countable choice for reals fails. Assuming the consistency of an inaccessible cardinal, "projective" can even be replaced with "$L(\mathbb{R})$". This vastly strengthens earlier consistency results in the literature.
